Abstract
By a curve in Rd we mean a continuous map γ: I -> Rd, where I is a subset of R is a closed interval. We call a curve γ in Rd (< k+1)-crossing if it intersects every hyperplane at most k times (counted with multiplicity).
The (< d+1)-crossing curves in Rd are often called convex curves and they form an important class; a primary example is the moment curve {(t, t2, ..., td): t in [0, 1]}. They are also closely related to Chebyshev systems, which is a notion of considerable importance, e.g., in approximation theory.
Our main result is that for every d there is M = M(d) such that every (< d + 2)-crossing curve in Rd can be subdivided into at most M(< d+1)-crossing curve segments. As a consequence, based on the work of Eliáš, Roldán, Safernová, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in Rd concerning order-type homogeneous sequences of points, investigated in several previous papers.